*3.1. Validation*

The standard benchmarking problem of natural convection due to the hot circular cylinder inside a square enclosure [36,49] is chosen to validate the numerical code developed based on the present simulation technique (see Figure 2 for simulation set-up). The following equation is considered to fix the kinematic viscosity, ν [37]

$$\nu = \sqrt{\frac{\text{Pr}}{\text{Ra}}} \mathcal{U}\_{\text{c}} L\_{\text{c}} \tag{13}$$

where Pr, *Uc*, and *Lc* are the Prandtl number, the characteristic velocity, and the characteristic length, respectively, and the definitions for Ra, Pr, and *Uc* are given by

$$\text{Ra} = \frac{\text{g}\beta\Lambda TL\_{\text{c}}^3}{\nu\alpha}, \quad \text{Pr} = \frac{\nu}{\alpha}, \text{ and } \mathcal{U}\_{\text{c}} = \sqrt{\text{g}\beta\Lambda TL\_{\text{c}}}.\tag{14}$$

Δ*T* in the above equation is the temperature difference between the inner circular cylinder (hot, *Tin* = *Th* ≡ 1) and outer square cavity (cold, *Tout* = *Th* ≡ 1). The computational domain is divided into 201 × 201 lattice grid points. The simulations are performed for, Pr = 0.71 (i.e., heat transferring medium is air), *Uc* = 0.1, and *Lc* = *Lout* (size of the outer square cavity). Three different combinations for Rayleigh number, Ra = 104, 105, and 106, and the aspect ratio, *AR* = *Lin*/*Lout* ≡ 0.2, 0.4, and 0.6 are considered. Figure 3 shows the isotherms (left side) and streamlines (right side) inside the enclosure when Ra = 10<sup>6</sup> and *AR* = 0.2. Two symmetrical vortices appear in the upper region of the enclosure, as the natural convection flow intensity is predominant in the upper region of the cavity due to Ra value is very high. The details of the surface-averaged Nusselt number, Nu, on the inner cylinder at all combinations of Ra and *AR* considered in the present work are provided in Table 1. The corresponding results obtained by the previous works [36,49] are also given in Table 1. Nu increases with Ra for all values of *AR*. The streamlines and the isotherms patterns, and Nu values of all Ra and *AR* combinations are in excellent agreemen<sup>t</sup> with the previous results [33,35]. After this validation, simulation of fluid flow and heat transfer due to natural convection inside the hexagonal annulus is performed and the corresponding results are discussed in the following section.

**Figure 2.** Set-up, consists of an annular space between an outer square enclosure (cold) and an inner circular cylinder (hot), considered for validating the present numerical method.

**Figure 3.** Isotherms (left side) and streamlines (right side) patterns inside the square enclosure when Rayleigh number is, Ra = 106, and aspect ratio, *AR* = 0.2.

**Table 1.** The details of the surface average Nusselt number, Nu, on the inner cylinder at different combinations of Ra and *AR*.


### *3.2. Natural Convection in the Concentric Hexagonal Annulus*

In this session, the results obtained by the simulation of the fluid flow and heat transfer in the annulus bounded by two horizontal concentric hexagonal cylinders are presented (set-up is shown in Figure 1). Simulations are performed for different values of *AR* and Ra, by varying *AR* in the range, *AR* = 0.2~0.6, and Ra in the range, Ra = 10<sup>3</sup> ∼ 106. The characteristic length in Equation (14) is set as, *Lc* = *Lout* (the size of the outer hexagon).

All the results obtained by the present simulation technique are compared with those given by commercial software, ANSYS-Fluent®. Fluent 18.2 is used to simulate a steady laminar flow and heat transfer inside a two-dimensional annular space bounded by two concentric hexagonal cylinders. The size of the outer cylinder is fixed at *Lout* = 212 m and that of the inner cylinder is varied as per the *AR*. The values for the temperatures at the walls of the inner and outer cylinder are set at *Tin* = 289 K and *Tout* = 288 K, respectively. Constant temperature and no-slip BC are used for heat transfer and fluid flow, respectively. The initial value for density is taken as ρ0 = 1.225 kg/m<sup>3</sup> (air density value at temperature 288 K) and the Boussinesq model is used to model the variation of the density as a function of temperature. Dry air properties at temperature 288 K are used to set the values for specific heat, viscosity, thermal conductivity, and thermal expansion coefficient. The value for the *y*− directional gravitational acceleration constant is varied corresponds to Ra. SIMPLE (Semi-Implicit Method for Pressure Linked Equation) scheme has opted for the pressure-velocity coupling. The governing equations are discretized using the least square cell-based method and a second-order upwind scheme

is chosen to solve momentum and energy equations. The Gauss–Seidal iterative method with default under-relaxation factors is selected to solve the system of algebraic equations. The convergence criterion for the residuals of all continuity, momentum, and energy equations is set as 10−9.

### 3.2.1. Streamlines and Isotherms Patterns Inside the Annulus

It is observed from the simulation results that irrespective of *AR* and Ra values, the fluid flow patterns (streamlines) and temperature contours (isotherms) are symmetrical about the vertical centerline of the annulus. Figure 4 shows the isotherms (left side) and streamlines (right side) pattern inside the annulus for the two values of *AR* = 0.2 (Figure 4a) and *AR* = 0.6 (Figure 4b), when Ra = 103. As Ra is very low, the strength of the buoyancy force (strength of the gravitational acceleration in this case) that causes the convective flow is very low. Therefore, the heat transfer process inside the annulus is mainly dominated by the conduction mode and the isotherms are very smooth (no distortion of isotherms takes place due to very weak fluid flow) and are almost concentric to the inner and outer hexagonal cylinders. Both isothermal and streamlines are symmetrical concerning the vertical as well as horizontal centerlines of the annulus. When *AR* = 0.2, the isotherms are almost circular and the spacing between them increases with the distance from the inner hexagon as the available space between inner and outer hexagons is more than that when *AR* = 0.6. On the other hand, when *AR* = 0.6, the isotherms in the vicinity of both inner and outer cylinders are in the form of the hexagon and the spacing between them is less as they ge<sup>t</sup> squeezed due to constricted space between inner and outer hexagons. The streamlines pattern for both *AR* = 0.2 and *AR* = 0.6 show that two symmetrical recirculating eddies (kidney-shaped cells) are formed inside the annulus and the location of cell centers is almost close to the horizontal centerline of the annulus as the fluid flow intensity in the upward direction is almost negligible because of very low Ra. When we observe Figure 4b carefully, we can see that there is slight penetration of the streamline into the solid edges, which is a slight drawback of the present scheme. The main reason for this phenomenon is that as SPM is a non-conforming-mesh method, the same grid system is used for the solid and fluid regions and simulations are also performed inside the solid regions (even though it is enough to consider the boundary e ffects on the hexagonal edges). However, performing the simulations inside the solid does not a ffect the flow field in the fluid domain and the overall behavior of the fluid flow and heat transfer is well captured.

Figure 5 shows the isotherms (left side) and streamlines (right side) pattern inside the annulus for the values of *AR* = 0.2 (Figure 5a) and *AR* = 0.6 (Figure 5b), and when Ra = 106. As the Ra is very high, the e ffect of buoyancy-driven flow is significant and hence the heat transfer in the upper region of the annulus is mainly dominated by the convection mode. Isotherms and streamlines are no longer symmetrical about the horizontal median of the annulus. For *AR* = 0.2, it is concluded from the isotherms and streamlines pattern that the fluid near the inner hexagonal cylinder surface gets heated and moves upwards along the upper inclined edges of the hexagon due to the buoyancy e ffect. Because of strong convection currents, a thermal plume is formed on the top of the inner cylinder and thermal boundary layer thickness at the top flat edge of the outer cylinder is very thin (indicated by close clustering of isothermal lines) as continuous impingement of fluid flow in the upper region of the annulus. The thermal boundary layer thickness at the bottom of the inner hexagonal cylinder is also found to be very low and the fluid temperature below the inner cylinder is almost uniform and is equal to that of the outer cylinder as heat transfer in this region is dominated by conduction. The centers of the symmetrical recirculating eddies are located well above the horizontal median as the fluid flow is dominant in the upper half of the annulus. The streamline pattern for *AR* = 0.2 also reveals that two symmetric secondary vortices are formed at the bottom wall of the outer cylinder due to the separation of the momentum boundary layer as a result of strong upward convective flow.

**Figure 4.** Isotherms (left side) and streamlines (right side) patterns inside the hexagonal annulus when the Rayleigh number is, Ra = 103, and for *AR* = 0.2 (**a**), and *AR* = 0.6 (**b**).

**Figure 5.** Isotherms (left side) and streamlines (right side) patterns inside the hexagonal annulus when the Rayleigh number is, Ra = 106, and for *AR* = 0.2 (**a**), and *AR* = 0.6 (**b**).

A completely different phenomenon is observed when *AR* = 0.6. Since there is limited available space for convection on the top of the inner cylinder, two separate thermal plumes (due to the buoyancy-driven fluid flow along the upper inclined edges of the hexagon) are formed along each upper corner of the hexagon. A third thermal plume is also seen on the top flat edge of the inner cylinder in the reverse direction as the uppermost corner of the inner hexagonal cylinder separates the fluid flow and generates two secondary vortices. The fluid flow separation phenomenon can be confirmed by noticing the two counter-rotating cells over the top flat edge of the inner cylinder from the streamline pattern of *AR* = 0.6. This type of flow separation phenomena at a high *AR* value was also observed by Raithby et al. [6], Bararnia et al. [30], Moutaouakil et al. [32], and Hu et al. [36], and even though their simulation domains were completely different from the present study.

To assess the capability of the present simulation method for predicting the behavior of natural convection flow in the concentric hexagonal annulus, the results obtained from the present method are compared with ANSYS-Fluent® results. Figure 6 shows the simulation results of isotherms and streamlines patterns obtained from Fluent for the values of *AR* = 0.2 (Figure 6a) and *AR* = 0.6 (Figure 6b), and the case when Ra = 106. By comparing the isotherms and streamlines patterns of Figures 5 and 6, we can say that the present simulation results are successfully reproduced the Fluent results.

**Figure 6.** Isotherms (left side) and streamlines (right side) patterns obtained from ANSYS-Fluent® software for the values of *AR* = 0.2 (**a**) and *AR* = 0.6 (**b**) when Ra = 106.
